$\text{ If }z=\frac{1}{3}+\frac{1.3}{3.6}+\frac{\mathrm{1.3.5}}{\mathrm{3.6.9}}+\dots \dots \dots \mathrm{\infty }\text{ then find value of }\frac{z^2+2z+100}{10}\text{ is equal to }$

$\begin{array}{l}{\left(1-\frac{2}{3}\right)}^{-\frac{1}{2}}=1+\frac{1}{2}\cdot \frac{2}{3}+\frac{\frac{1}{2}\cdot \frac{3}{2}}{\lfloor 2}\cdot {\left(\frac{2}{3}\right)}^2+\frac{\frac{1}{2}\cdot \frac{3}{2}\cdot \frac{5}{2}{\left(\frac{2}{3}\right)}^3}{\frac{3}{2}}\\ \Rightarrow {\left(\frac{1}{3}\right)}^{-\frac{1}{2}}-1=\frac{1}{3}+\frac{1.3}{(3{)}^2\lfloor 2}+\frac{\mathrm{1.3.5}}{(3{)}^3\lfloor 3}+\dots \dots \dots \infty \\ \Rightarrow \sqrt{3}-1=z\Rightarrow z^2+2z-2=0\\ \Rightarrow z^2+2z=2\\ \end{array}$